Predictive Mean Matching using a Factor Model ,

Predictive Mean Matching using a Factor Model, Varriale - Guarnera – Nuremberg, 09/09/2013

Predictive Mean Matching using a Factor Model, an application to the Business Multipurpose Survey

Roberta Varriale, Ugo GuarneraIstat

Nuremberg, 09/09/2013


Sample survey on businesses carried out by Istat together with the Business and Services Census 2011

Features

Aims: identifying specific business profilesComplex questionnaireLarge number of variables (different nature)

Nonresponse-rate of 72k enterprises (more than 20 employees): 15% Difficulty to use an explicit parametric method for imputation

Nearest Neighbor Donor (NND): consists in matching completely observed units (donors) with incomplete units (recipients), based on some distance function, and transferring values from donors to recipients

Predictive Mean Matching (PMM): distance function “weights” the covariates used in NND with its relative predictive power with respect to the “target” variables

Motivation: (Business) Multi Purpose Survey


Predictive Mean Matching


Y = Y1,…,YP variables of a sample survey to be imputed X = X1,…,XQ variables available for all units (covariates)

Y continuous, a typical application of PMM:

1.the parameters of the regression model of Y on X are estimated with standard methods

2.based on the estimates from step 1, predictive means Y*≡ E (Y | X) are computed both for the units with missing values (recipients) and units with complete data (donors)

3.for each receiver ur a donor ud is selected in order to minimize the Mahalanobis distance defined by:

D(ud, ur) ≡ (yd* - yr*)T S-1 (yd* - yr*)

where yd* and yr* are the predictive means estimates on donor and recipient, respectively, and S is the residual variance-covariance matrix of the regression model

4.each ur is imputed by transferring the Y values from its closest donor



PMM

Mahalanobis distance(residual covariance matrix from the regression model)

Research problem:

Which model and which distance function has to be used when the target variables are not all continuous?

MPS: target variables are all categorical

Titolo intervento, nome cognome relatore – Luogo, data


“weights” the covariates to be used in NND with its relative predictive power with respect to the target variables

establishes the distance between units weighting the “goodness of prediction” of the single target variables largest “weights” to the target variables of the predictive means with the smallest prediction error


In PMM for categorical variables to be imputed,

the “natural” imputation model to define an appropriate distance is the log-linear model with covariates X

distance function on the estimated probabilities of the categories of the target variables

2 main limitations:

1.the model is very complex when the number of variables used in the imputation model is big (i.e. only when we are able to set up and process the full multi-way cross-tabulation required for the log-linear analysis)

2.the distance function has to take into account both the distance between the expected frequencies of the multi-way cross-tabulation and the variability due to the estimation process


PMM with latent variables (factor model)


The model is composed of 2 parts:1.factor model links the latent factor to the observed indicators 2.regression model links the covariates to the latent factor

Structural Equation ModelMIMIC (Multiple Indicators Multiple Causes) model

Factor model (with covariates)

y1

yP

yp

…

…

η

x1

xQ

xq

…

…

u



y1

yP

yp

…

…

η

x1

xQ

xq

…

…

u


vp= µp + λpη

gp (E(yp|η))= vp



y1

yP

yp

…

…

η

x1

xQ

xq

…

…

u


η= α0+ α1X1 + … + αQXQ + u




Y = Y1,…,YP variables of a sample survey to be imputed X = X1,…,XQ variables available for all units (covariates)

Y categorical (or not all continuous), the PMM with a latent factor:

1.the parameters of the factor model with covariates are estimated with standard methods

2.based on the estimates from step 1, predictive means η*≡ E (η | X) are computed both for the units with missing values (recipients) and units with complete data (donors)

3.for each receiver ur a donor ud is selected in order to minimize the distance between η values

4.each ur is imputed by transferring the Y values from its closest donor

Titolo intervento, nome cognome relatore – Luogo, data

PMM with one latent factor

The “matching variable” of PMM:

(expected value of) the latent factor η

summarizes data variability (response variables yp )

is related to the variables X1,..,XQ


Target variables: Y1: type and nationality of decision management (4 categories)Y2: employees with high skills (2 categories)Y3: type of partnership (3 categories)Y4: delocalization of specific production functions (2 categories)

Covariate:X1: number of employees (continuous)X2: added values (continuous)X3: turnover (continuous)X4: membership in an enterprise group (2 categories)

Stratification variables in the imputation process: Section of economic activity (ATECO)Export/import activity

(Business) Multi Purpose Survey

• 0=other• 1=physical person/family, Itaian• 2= other firm, Italian• 3=abroad

• 0=no partnership• 1= both formal and informal partn.• 2= only formal partnership

yes/no

High/not high


Monte Carlo simulation study

Data MPS, available at 20/12/2012

# enterprises: 39826 (20 or more than 20 employees)

200 iterations

1.Simulation of the item nonresponse (20% of the total number of observations) on Y variables according to a Missing at Random (MAR) mechanism

2.Estimation of the marginal and joint frequencies of the categories of Y1,…,Y4 for the dropped units, using different methods

3.Evaluation of the different methods by comparing the true and estimated frequencies obtained at each iteration with the Hellinger distance, and averaging the results over the 200 replications (frequencies are compared separately for each estimation domain defined by the ATECO)

Simulation study

Software: R, Latent Gold


2. Estimation of the marginal and joint frequencies of the categories of Y1,…,Y4 for the dropped units, using different methods

Simulation study



Computation of the expected frequencies according to the estimates obtained by the models:

Logit: multinomial logit model

Factor: factor model with covariates

Simulation study






Draw realization of Y1,…,Y4 using the probabilities estimated by the models:

Logit.Rnd: multinomial logit model

Factor.Rnd: factor model with covariates

Simulation study






Draw realization of Y1,…,Y4 using the probabilities estimated by the models:

Logit.Rnd: multinomial logit model

Factor.Rnd: factor model with covariates

Imputation method NND (Euclidean distance):

X.Donor, matching variables: X1,…,X4

Logit.Donor, matching variables: probabilities of each category of Y1,…,Y4 estimated by a multinomial logit model with covariates X1,…,X4

Factor.Donor, matching variable: value of the latent factor estimated by a factor model with covariates (PMM with a latent factor)

Simulation study


The Hellinger distance computed for each method is similar

The reduction of dimensionality performed by the PMM with a factor model does not harm the results of the imputation process

The performance of the NND methods is very similar to that ones of the corresponding methods based on directly drawing from the estimated probability of the categories of Y an advantage of using a NND is that it allows us to impute all variables of each incomplete record, rather than only the target variables

The additional variability introduced by the random drawing methods result in a small increase of the Hellinger distance values

Simulation study - results




















The results show a quite good performance of the PMM with a factor model

Further analysis:(additional) analysis of the nonresponse process (nonresponse pattern)(additional) analysis of the variable “meanings”

Monte Carlo experiments using simulated data

Suggestions?

Thank you for your [email protected], [email protected]

Future research

subjectmatter

methodological matter


References

Bartholomew, D.J., Knott, M. (1999). Latent variable models and factor analysis. Arnold, London

Bollen, K.A. (1989). Structural equations with latent variables. J.Wiley, New York

Di Zio, M., Guarnera, U. (2009). Semiparametric predictive mean matching. AStA - Advances in Statistical Analysis. 93, 175-186

Little, R.J.A. (1988). Missing-data adjustments in large surveys. Journal of Business & Economic Statistics. 6, 287-296

Skrondal, A., Rabe-Hesketh, S. (2004). Generalized latent variables modeling: multilevel, longitudinal, and structural equation models. Boca Raton, FL: Chapman & Hall/CRC

Lombardi, S., Lorenzini F., Verrecchia F. (2012). Three Pillars for a New Statistical System on Enterprises: Business Register, Thematic Surveys and Business Census 2011. In: Fourth International Conference on Establishment Surveys (ICES-IV), Montreal, Canada, 11-14 June

Vermunt, J.K., van Ginkel, J.R., van der Ark, L.A., Sijtsma K. (2008). Multiple imputation of incomplete categorical data using latent class analysis. Sociological Methodology 33, 369–297

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Predictive Mean Matching using a Factor Model ,